
In this manuscript, we consider a preconditioned metric which is extended from the matrix case [1]}, [2]}, [3]}, [4]}.
Let \(\mathbb {R}_{\ast }^{n_1\times r}\times \mathbb {R}_{\ast }^{n_2\times r}\)  be the space of matrices with full column rank \(r\) . Given \(\bar{x} := \left\lbrace \mathbf {G},\mathbf {H}\right\rbrace \in \mathbb {R}_{\ast }^{n_1\times r}\times \mathbb {R}_{\ast }^{n_2\times r}\) , the preconditioned metric on the tangent space of \(\mathbb {R}_{\ast }^{n_1\times r}\times \mathbb {R}_{\ast }^{n_2\times r}\)  at \(\bar{x}\)  is defined as
\(\bar{g}_{\bar{x}}\left(\bar{\xi },\bar{\eta }\right)= \operatorname{trace}\left({\xi ^1}^\mathsf {T}\eta ^1\left(\mathbf {H}^\mathsf {T}\mathbf {H}\right)\right)+\operatorname{trace}\left({\xi ^2}^\mathsf {T}\eta ^2\left(\mathbf {G}^\mathsf {T}\mathbf {G}\right)\right),\) 
